A Gaussian wave packet at time

can be represented by

,

where

is the initial width of the wave packet and

is the momentum.

At a later time

, the wave packet evolves with

,

where

is a constant,

is the mass of the particle, and

is a parameter that determines the corresponding width of the wave packet (

is the actual width). The parameter

is given by

. Clearly, the width increases with time

, as the wave packet spreads. For simplicity,

and

have been set equal to unity.

The probability amplitude is a Gaussian function centered about the point

.

The wave packet maintains a Gaussian shape, with changing centroid and width.

Snapshot 1: small initial width implies faster spread

Snapshot 2: initially broad wave packet spreads out more slowly

In the limiting case, of a wave packet initially equal to a Dirac delta function, it immediately transforms into an infinite plane wave.

[1] H. C. Verma,

*Quantum Physics*, 2nd ed., Bhopura, Ghaziabad, India: Surya Publications, 2009.